Sign in
Please select an account to continue using cracku.in
↓ →
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.
Correct Answer: 1086
We need to find the total number of four-digit numbers such that each of the first three digits is divisible by the last digit.
Let the four-digit number be $$\overline{abcd}$$ where $$a$$ is the thousands digit, $$b$$ the hundreds, $$c$$ the tens, and $$d$$ the units digit. The conditions are $$d | a$$, $$d | b$$, $$d | c$$, $$a \geq 1$$, and $$d \geq 1$$.
d = 1: Any digit is divisible by 1, so $$a \in \{1,...,9\}$$ (9 choices), $$b, c \in \{0,...,9\}$$ (10 choices each), giving $$9 \times 10 \times 10 = 900$$.
d = 2: $$a \in \{2,4,6,8\}$$ (4 choices), $$b, c \in \{0,2,4,6,8\}$$ (5 choices each), giving $$4 \times 5 \times 5 = 100$$.
d = 3: $$a \in \{3,6,9\}$$ (3 choices), $$b, c \in \{0,3,6,9\}$$ (4 choices each), giving $$3 \times 4 \times 4 = 48$$.
d = 4: $$a \in \{4,8\}$$ (2 choices), $$b, c \in \{0,4,8\}$$ (3 choices each), giving $$2 \times 3 \times 3 = 18$$.
d = 5: $$a \in \{5\}$$ (1 choice), $$b, c \in \{0,5\}$$ (2 choices each), giving $$1 \times 2 \times 2 = 4$$.
d = 6: $$a \in \{6\}$$ (1 choice), $$b, c \in \{0,6\}$$ (2 choices each), giving $$1 \times 2 \times 2 = 4$$.
d = 7: $$a \in \{7\}$$ (1 choice), $$b, c \in \{0,7\}$$ (2 choices each), giving $$1 \times 2 \times 2 = 4$$.
d = 8: $$a \in \{8\}$$ (1 choice), $$b, c \in \{0,8\}$$ (2 choices each), giving $$1 \times 2 \times 2 = 4$$.
d = 9: $$a \in \{9\}$$ (1 choice), $$b, c \in \{0,9\}$$ (2 choices each), giving $$1 \times 2 \times 2 = 4$$.
$$900 + 100 + 48 + 18 + 4 + 4 + 4 + 4 + 4 = 1086$$ Therefore, the answer is $$\textbf{1086}$$.
Create a FREE account and get:
Predict your JEE Main percentile, rank & performance in seconds
Educational materials for JEE preparation
Ask our AI anything
AI can make mistakes. Please verify important information.
AI can make mistakes. Please verify important information.